Abstract
We revisit Heisenberg indeterminacy principle in the light of the Galois–Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois–Grothendieck duality between finite K-algebras split by a Galois extension \(L\) and finite \(Gal(L{:}K)\)-sets can be reformulated as a Pontryagin duality between two abelian groups. We define a Galoisian quantum model in which the Heisenberg indeterminacy principle (formulated in terms of the notion of entropic indeterminacy) can be understood as a manifestation of a Galoisian duality: the larger the group of automorphisms \(H\subseteq G\) of the states in a G-set \({\mathcal {O}}\simeq G/H\), the smaller the “conjugate” algebra of observables that can be consistently evaluated on such states. Finally, we argue that states endowed with a group of automorphisms \(H\) can be interpreted as squeezed coherent states, i.e. as states that minimize the Heisenberg indeterminacy relations.
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Notes
For a recent version of the epistemic view of quantum mechanics see for instance [18].
The term observable algebra denotes the algebra of functions on a space \(M\). Roughly speaking, the evaluation of the functions in the observable algebra on the points of \(M\) can be interpreted as “observations” whose numerical results allow us to discern (or separate) its different points.
It is worth noting that the L-spectrum of \(\mathcal {B}\otimes _{K}L\), that is \(Spec_{L}(\mathcal {B}\otimes _{K}L)=Hom_{L\text{- }Alg}(\mathcal {B}\otimes _{K}L, L)\), coincides with \(Spec_{K}(\mathcal {B})\) [4].
The semilinearity of the action means that \(g\cdot (la\,+\,bb')= g(l)g(a) +(g\cdot b) (g\cdot b')\) for all \(l\in L\) and all \(a, b, b'\) in \({\mathcal {A}}\).
Indeed, \(Hom_{Group}(G, (L^{*}, \times ))\cong Hom_{Group}(G, (\mu _n(L), \times )) \cong Hom_{Group}(G, (\mu _n({\mathbb {C}}), \times )) \cong \widehat{G}\). This results from the fact that \(\chi (g)^n = \chi (g^n) = \chi (1_G)=1_L\) for every \(\chi \in Hom_{Group}(G, (L^{*}, \times ))\) and every \(g\in G\). Therefore, \(Im \chi \subseteq \mu _n(L)= \mu _n({\mathbb {C}})\).
For instance, the inverse (discrete) Fourier transform can be defined by
$$\begin{aligned} \mathcal {F}^{-1} : L[\widehat{G}]&\rightarrow L^G\\ f\doteq \sum _{\chi \in \widehat{G}} f(\chi )\chi&\mapsto \mathcal {F}^{-1}(f), \end{aligned}$$where
$$\begin{aligned} \mathcal {F}^{-1}(f)(g) \doteq \frac{1}{\sqrt{n}} \sum _{\chi \in \widehat{G}} f(\chi )\chi (g), \end{aligned}$$for \(n=Card(G)\). This expression explicitly shows that this inverse Fourier transform is just the linear extension of the Gelfand transform between \(\widehat{G}\) and the elementary observables on \(G\).
The isomorphism is given by \(\chi \in H^{\perp } \mapsto \widetilde{\chi }\in \widehat{G/H}\) such that \(\widetilde{\chi }([g])=\chi (g)\).
Using (4), we have that \(\chi (g)=1\) for all \(\chi \in H^{\perp }\) iff \(\widetilde{\chi }([g])=1\) for all \(\widetilde{\chi } \in \widehat{G/H}\). Now this is the case iff \([g]=1\) in \(G/H\), that is if \(g \in H\).
These conditions are a combination of classical results related to the abelian inverse Galois problem, the cyclotomic extensions, and the classical Galois theory (see [13, pp. 266–268]).
For instance, in [21] the author analyzes quantum models in which the position and the momentum do not take values in \(\mathbb {R}\), but rather in the so-called Galois fields \(\mathbb {Z}_{p^{n}}\) (the integers modulo a positive integer power of a prime number \(p\)). In [10], the authors use Galois fields as a substitute for \(\mathbb {C}\) in the Hilbert space structure.
In [14], Majid proposes a similar analysis of the “Fourier-Pontryagin duality” between a structure and the family of “representations” that we need in order to reconstruct it.
Some authors argue that the notion of entropic indeterminacy encodes the quantum indeterminacy in a more efficient way than the standard deviations \(\Delta x\) and \(\Delta p\) (see [3]).
This is the continuous analog of the Shannon entropy associated to a discrete random variable \(X\) with possible values \(\{x_1, ..., x_n\}\) and discrete probability distribution \(P(X)\). Indeed, the Shannon entropy is defined by the expression \(H(X) \doteq - \sum _{1\le i \le n} {P(x_i) ln P(x_i)}\). In the continuous case \(G=\mathbb {R}\), \(\mid f\mid ^2\) is interpreted as the probability density defined by a wave function \(f\in L^2(G,{\mathbb {C}})\).
In the continuous case, the unitary dual of \(G=\mathbb {R}^m\) is \(\widehat{G} \cong \mathbb {R}^m\). A particular value of the “momentum” \(\chi \in \mathbb {R}^m \cong \widehat{G}\) defines the elementary observable on \(G\) given by \(\tilde{\chi } : g \mapsto e^{i \chi \cdot g}\). The discrete case is analogous. The unitary dual of \(G=\mathbb {Z}/n\mathbb {Z}\) is given by \(\widehat{G} \cong \mathbb {Z}/n\mathbb {Z}\). A particular value of the “momentum” \(\chi \in \mathbb {Z}/n\mathbb {Z} \cong \widehat{G}\) defines the elementary observable on \(G\) given by \(\tilde{\chi } : g \mapsto e^{\frac{2i\pi \chi g}{n}}\).
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Acknowledgments
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013 Grant Agreement n\(^{\circ }\) 263523). We also thank Daniel Bennequin, Mathieu Anel, and Alexandre Afgoustidis for helpful discussions.
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Page, J., Catren, G. Towards a Galoisian lnterpretation of Heisenberg lndeterminacy Principle. Found Phys 44, 1289–1301 (2014). https://doi.org/10.1007/s10701-014-9812-2
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DOI: https://doi.org/10.1007/s10701-014-9812-2